Operational and Algebraic Aspects of Appell–Hermite–Fibonacci Polynomials in F-Golden Calculus and Their Applications

In this paper, we introduce the Appell–Hermite–Fibonacci polynomials within the framework of Fibonomial calculus, combining the Appell structure with a Hermite-type deformation governed by Fibonacci coefficients. The family is defined through an appropriate F-exponential generating function, from which its principal properties naturally follow. Explicit representations, including convolution identities and series expansions, are established. A determinantal formulation preserving the lower–Hessenberg structure of Fibonacci–Appell systems is obtained, together with a matrix realization linked to a generalized Fibonacci–Pascal matrix. The operational framework yields lowering and raising relations, a second-order F-differential equation, and a Rodrigues-type representation. Under suitable conditions, orthogonality properties are also derived. The results place this hybrid family within a coherent extension of Appell theory in the Fibonacci setting.